Functions: Composition, Piecewise Definitions, and Transformations

Functions and Their Properties

Introduction to Functions

Functions are fundamental objects in mathematics that assign each input exactly one output. Understanding how to manipulate, compose, and graph functions is essential for success in Precalculus and Calculus.

Function Composition

Definition and Notation

Function composition involves combining two functions so that the output of one function becomes the input of another. If f and g are functions, the composition is written as g ˆ f or g(f(x)).

• Domain of f: The set of all possible inputs for f.

• Range of f: The set of all possible outputs for f, which becomes the domain for g in the composition.

• Domain of g: The set of all possible inputs for g.

• Range of g: The set of all possible outputs for g.

Formula:

$g \circ f(x) = g(f(x))$

Example: If $f(x) = 3x^2$ and $g(x) = \frac{1}{2}x + \frac{3}{2}$, then:

• First, compute $f(3) = 3 \times 3^2 = 27$

• Then, $g(f(3)) = g(27) = \frac{1}{2} \times 27 + \frac{3}{2} = \frac{27}{2} + \frac{3}{2} = 15$

Piecewise Functions

Definition and Structure

Piecewise functions are defined by different expressions depending on the subset of the domain. They are useful for modeling situations where a rule changes based on the input value.

General Form:

$f(x) = \begin{cases} \text{expression 1} & \text{if condition 1} \\ \text{expression 2} & \text{if condition 2} \end{cases}$

• Each 'piece' of the function applies to a specific interval or condition.

• The absolute value function $y = |x|$ is a classic example of a piecewise function.

Example: Tax Brackets

Tax systems often use piecewise functions to calculate tax owed based on income brackets. For example, a function d(x) might be defined differently for various ranges of x (income).

Graphical Representation: Piecewise functions can be graphed by plotting each piece over its respective domain interval.

Example Problem: y-intercept of a Piecewise Function

Given:

$f(x) = \begin{cases} 4 - x^2 & x \neq 0 \\ 3 & x = 0 \end{cases}$

• The y-intercept occurs at $x = 0$.

• Thus, $f(0) = 3$.

• Answer: The y-intercept is 3.

Transformations of Functions

Types of Transformations

Transformations change the appearance of a function's graph without altering its basic shape. Common transformations include:

• Translations: Shifting the graph horizontally or vertically.

• Reflections: Flipping the graph over a line (such as the x-axis or y-axis).

• Stretches and Compressions: Expanding or contracting the graph vertically or horizontally.

Example: The graph of $f(x) = x^2$ shifted up by 3 units is $f(x) = x^2 + 3$.

Summary

• Function composition allows for combining functions to create new ones.

• Piecewise functions are defined by different rules over different parts of their domain.

• Transformations help in graphing and understanding how functions behave under shifts, reflections, and stretches.